Weakly rarefied gas flows between nonuniformly heated parallel plates

The aim of this research is to establish the validity of the predictions of the theory of slow nonisothermal flows, to study the limits of applicability (with respect to the Knudsen number) of the conclusions reached and to determine the effect of the Knudsen layers on these flows on the basis of a numerical investigation of slow nonisothermal weakly rarefied gas flow in a plane infinite channel with weakly nonequilibrium heating of the walls and a finite wall temperature difference. The gas flow is described by a relaxation transport equation. The results obtained show how quickly, as the Knudsen number decreases, the solutions of the transport equation outside the Knudsen layers tend to the solution of the equations of gas dynamics of slow nonisothermal flows (and not to the solution of the Navier-Stokes equations).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 115–121, January–February, 1988.     

Slow rotation of a sphere in a confined volume of a dilute gas

The axially symmetric motion of a gas in a volume confined between an external immobile surface of rotation and a coaxial surface of a rotating sphere is considered. A solution is obtained by the moment method based on the Boltzmann equation with a collision integral of Maxwellian molecules. The gas-velocity distribution and an expression for the friction torque exerted on the sphere are obtained for arbitrary Knudsen numbers and for an arbitrary shape of the outer surface. The proportionality of the gas slip velocity over the surface of the sphere to the friction strain is shown. The friction torque is investigated for specific shapes of the outer surface. The motion of a gas filling the space between concentric spheres, each of which rotates about an arbitrary axis, is treated. In the limiting case of small Knudsen numbers the expressions obtained are compared with the corresponding results for a continuous medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117–124, July–August, 1978.The authors are grateful to the participants of the seminars guided by G. I. Petrov and A. M. Golovnyi for discussions concerning this work.     

Macroscopic equations of motion of disperse systems

The macroscopic equations of motion of a two-component system consisting of a continuous phase and a large number of solid particles are considered. The generalized kinetic equation of a pseudogas obtained earlier by the author is expressed in a form more convenient for calculations. The Chapman-Enskog method is used to solve the kinetic equation at small Knudsen numbers and dimensionless number characterizing the transfer of momentum between the phases of order unity. Because of the influence of the continuous phase, the stress tensor in the macroscopic conservation equations of the pseudogas is anisotropic. The obtained macroscopic equations of the pseudogas are more general than the ones proposed earlier by Myasnikov, this being due to the anisotropy of the time constants which occur in the operator of the hydrodynamic interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 39–44, March–April, 1980.I thank V. P. Myasnikov for posing the problem and for helpful discussions.     

Diffusion baroeffect in a capillary in a broad range of knudsen numbers

The magnitude of the diffusion baroeffect in a capillary is calculated in the entire range of Knudsen numbers based on a solution of BGK equations for a binary gas equation. The theoretical magnitudes of the baroeffect agree well with the experimental results obtained in a broad range of Knudsen numbers. The thermodynamic coefficients are calculated, and the Onsager reciprocal relation is demonstrated, from which in the limit of a viscous system it follows that the barodiffusion constant is equal to the coefficient of diffusion slippage.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 74–79, March–April, 1973.     

Analysis of the Onsager Relations by Analytic Methods in Kinetic Gas Theory

The well-known Onsager relations L ₁₂=L ₂₁ are verified within the framework of kinetic gas theory with allowance for the mass and heat fluxes localized in the Knudsen layer. On the basis of an analytic solution of the BGK (Bhatnagar-Gross-Krook) equation, it is shown that the Onsager relations are fulfilled correct to at least exponential corrections (–1/Kn) in the Knudsen number Kn.     

Homogenization of Wall-Slip Gas Flow Through Porous Media

The permeability of reservoir rocks is most commonly measured with an atmospheric gas. Permeability is greater for a gas than for a liquid. The Klinkenberg equation gives a semi-empirical relation between the liquid and gas permeabilities. In this paper, the wall-slip gas flow problem is homogenized. This problem is described by the steady state, low velocity Navier–Stokes equations for a compressible gas with a small Knudsen number. Darcy's law with a permeability tensor equal to that of liquid flow is shown to be valid to the lowest order. The lowest order wall-slip correction is a local tensorial form of the Klinkenberg equation. The Klinkenberg permeability is a positive tensor. It is in general not symmetric, but may under some conditions, which we specify, be symmetric. Our result reduces to the Klinkenberg equation for constant viscosity gas flow in isotropic media.     

An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit

In this article, we propose a numerical scheme to solve the Kac model of the Boltzmann equation for multiscale rarefied gas dynamics. Formally, this scheme is shown to be uniformly stable with respect to the Knudsen number, consistent with the fluid-diffusion limit for small Knudsen numbers, and with the Kac equation in the kinetic regime. Our approach is based on the micro–macro decomposition which leads to an equivalent formulation of the Kac model that couples a kinetic equation with macroscopic ones. This method is validated with various test cases and compared to other standard methods.     

Strong recondensation in a one-component and a two-component rarefied gas for an arbitrary value of Knudsen number

As is known, surface phenomena such as evaporation, absorption, and reflection of molecules from the surface of a body depend strongly on its temperature [1–5]. This leads to the establishment of a flow of a substance between two surfaces maintained at different temperatures (recondensation). The phenomenon of recondensation was studied in kinetic theory comparatively long ago. However, up to the present, only the case of small mass flows in a onecomponent gas has been investigated completely [3,4]. Meanwhile it is clear that by the creation of appropriate conditions we can obtain considerable flows of the recondensing substance, so that the mass-transfer rate will be of the order of the molecular thermal velocity. Such a numerical solution of the problem with strong mass flows along the normal to the surface for small Knudsen numbers for a model Boltzmann kinetic equation was obtained in [7]. In this study we numerically solve the problem of strong recondensation between two infinite parallel plates over a wide range of Knudsen numbers for a one-component and a two-component gas, on the basis of the model Boltzmann kinetic equation [6] for a one-component gas and the model Boltzmann kinetic equation for a binary mixture in the form assumed by Hamel [8], for a ratio of the plate temperatures equal to ten. We also investigate the effect of the relative plate motion on the recondensation flow.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1972.     

Diffusiophoresis of an aerosol particle in a binary gas mixture

The diffusion force and rate are calculated for the diffusiophoresis of a spherical particle in a binary gas mixture by solving the gas–kinetic equations. Two schemes of diffusiophoresis are considered: constant–pressure diffusion and diffusion of one mixture component through the other fixed component. The problem is solved by the integral–momentum method at arbitrary Knudsen numbers. Diffuse scattering of the gas molecules on the particle surface is assumed. The Lorentzian and Rayleigh models of a binary gas mixture are considered. The dependences of the force and rate of diffusiophoresis on the Knudsen number and the other determining parameters are analyzed. The results obtained are compared with well–known experimental data.     

From Boltzmann’s Equation to the Incompressible Navier–Stokes–Fourier System with Long-Range Interactions

We establish a rigorous demonstration of the hydrodynamic convergence of the Boltzmann equation towards a Navier–Stokes–Fourier system under the presence of long-range interactions. This convergence is obtained by letting the Knudsen number tend to zero and has been known to hold, at least formally, for decades. It is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered. However, these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption, which requires that the cross sections be integrable. Indeed, as soon as long-range intermolecular forces are present, non-integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas. In this long-range setting, the Boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case. In fact, the DiPerna–Lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the Boltzmann equation are the so-called renormalized solutions with a defect measure developed by Alexandre and Villani. Our work overcomes the new mathematical difficulties coming from the consideration of long-range interactions by proving the hydrodynamic convergence of the Alexandre–Villani solutions towards the Leray solutions.     

Degeneracy of Chapman-Enskog series for the transport properties in the case of slow steady flows of weakly rarefied gases

The structure of the Chapman-Enskog solution of the Boltzmann equation linearized with respect to the absolute Maxwell equilibrium is studied. Under the assumption of uniqueness and existence of a solution it is shown that in the steady case the series describing the transport phenomena consist of a finite number of terms, and the heat fluxes and diffusion rates are given by the Burnett approximation and the stresses by the super-Burnett approximation, the following terms of the series vanishing. At the same time, the gas-dynamic variables in all approximations in the small Knudsen number K satisfy the conservation equations in the Stokes approximation; the forces and moments acting on bodies placed in a mixture of gases can be calculated from the Navier-Stokes stresses without allowance for their reprocessing in Knudsen layers. A problem is formulated for a simple gas, and the transport properties are analyzed by using the invariance properties of the linearized Boltzmann equation and by means of the algorithm of the Chapman-Enskog method, and then the results are generalized to a mixture of gases, and the question of the forces and moments is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 157–163, July–August, 1988.I thank M. N. Gaidukov and O. G. Fridlender for fruitful discussions.     

Influence of excitation of internal degrees of freedom of molecules on the process of weak evaporation or condensation

A study is made of the process of weak evaporation (or condensation) with allowance for excitation of vibrational and rotational degrees of freedom of diatomic molecules. The solution to the corresponding Knudsen layer problem is obtained on the basis of a model kinetic equation of the type of the Morse equation [1]. A relation is obtained that establishes the connection between the rate of evaporation (or condensation) and the parameters of the surface and the gas above it. The boundary conditions of slip for the equations of gas dynamics are analyzed. The results are compared with the evaporation or condensation in the case of a monatomic gas. The introduction of accommodation coefficients for an evaporating surface is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 98–110, November–December, 1979.     

A method for deriving hydrodynamic equations for complicated systems

A general algorithm is proposed for constructing a uniformly valid asymptotic solution of the kinetic equations under conditions when the number of slowly varying macroscopic variables is greater than the number of integral invariants of the collision operator. The case of a chemically reacting gas mixture is considered, and a method for constructing the asymptotic solution for this case is described. The hydrodynamic equations for reacting and relaxing gas mixtures are described in general form and it is noted that consistent allowance for the disequilibrium of the reaction and relaxation processes leads to the appearance in the hydrodynamic equations of a number of additional terms, which describe the dependence of the rates of these processes on the spatial derivatives of the hydrodynamic variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 96–105, May–June, 1981.     

Intense evaporation of a gas from a two-dimensional periodic surface

Evaporation (or condensation) of a gas is said to be intense when the normal component of the velocity of the gas in the Knudsen layer has a value of the order of the thermal velocity of a molecule, c_T=(2kT/m)^1/2. In this case the distribution function of the molecules with respect to their velocities in the Knudsen layer differs from the equilibrium (Maxwellian) value by its own magnitude. As a result of this, over the thickness of the Knudsen layer the macroparameters also vary by their own magnitudes. So in order to obtain the correct boundary conditions for the Euler gas dynamic equations, it is necessary to solve the nonlinear Boltzmann equation in the Knudsen layer. The problem of obtaining such boundary conditions for the case of a plane surface was considered in [1–11]. In the present study this problem is solved for a two-dimensional periodic surface in the case when the dimensions of the inhomogeneities are of the order of the mean free path of the molecules and the inhomogeneities have a rectangular shape. The flow in the Knudsen layer becomes two-dimensional, and this leads to a considerable complication of the solution of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 132–139, March–April, 1985.In conclusion the author would like to express his gratitude to V. A. Zharov for his valuable advice, and also V. S. Galkin, M. N. Kogan, and N. K. Makashev for discussion of the results obtained.     

Asymptotic Solutions of the Equations for a Viscous Heat-Conducting Compressible Medium

Small nonstationary perturbations in a viscous heat-conducting compressible medium are analyzed on the basis of the linearization of the complete system of hydrodynamic equations for small Knudsen numbers (Kn ≪ 1). It is shown that the density and temperature perturbations (elastic perturbations) satisfy the same wave equation which is an asymptotic limit of the hydrodynamic equations far from the inhomogeneity regions of the medium (rigid, elastic or fluid boundaries) as M _a = v/a → 0, where v is the perturbed velocity and a is the adiabatic speed of sound. The solutions of the new equation satisfy the first and second laws of thermodynamics and are valid up to the frequencies determined by the applicability limits of continuum models. Fundamental solutions of the equation are obtained and analyzed. The boundary conditions are formulated and the problem of the interaction of a spherical elastic harmonic wave with an infinite flat surface is solved. Important physical effects which cannot be described within the framework of the ideal fluid model are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 76–87.Original Russian Text Copyright © 2005 by Stolyarov.     

气体动理学格式研究进展

介绍了近年来气体动理学格式(gas-kinetic scheme, GKS, 亦简称BGK 格式) 的主要研究进展, 重点是高阶精度动理学格式及适合从连续流到稀薄流全流域的统一动理学格式. 通过对速度分布函数的高阶展开和对初值的高阶重构, 构造了时间和空间均为三阶精度的气体动理学格式. 研究表明, 相比于传统的基于Riemann 解的高阶格式, 新格式不仅考虑了网格单元界面上物理量的高阶重构, 而且在初始场的演化阶段耦合了流体的对流和黏性扩散, 也能够保证解的高阶精度. 该研究为高精度计算流体力学(computatial uiddymamics, CFD) 格式的建立提供了一条新的途径. 通过分子离散速度空间直接求解Boltzmann 模型方程,在每个时间步长内将宏观量的更新和微观气体分布函数的更新紧密地耦合在一起, 建立了适合任意Knudsen(kn) 数的统一格式, 相比于已有的直接离散格式具有更高的求解效率. 最后, 本文还讨论了合理的物理模型对数值方法的重要性. 气体动理学方法的良好性能来自于Boltzmann 模型方程对计算网格单元界面上初始间断的时间演化的准确描述. 气体自由运动与碰撞过程的耦合是十分必要的. 通过分析数值激波层内的耗散机制,我们认识到采用Euler 方程的精确Riemann 解作为现代可压缩CFD 方法的基础具有根本的缺陷, 高马赫数下的激波失稳现象不可避免. 气体动理学格式为构造数值激波结构提供了一个重要的可供参考的物理机制.      

Solution of the temperature jump (Knudsen layer flow) and linear heat-transfer problems for two parallel plates in a rarefied gas

The Monte Carlo method [1, 2] is used to solve the linearized Boltzmann equation for the problem of heat transfer between parallel plates with a wall temperature jump (Knudsen layer flow). The linear Couette problem can be separated into two problems: the problem of pure shear and the problem of heat transfer between two parallel plates. The Knudsen layer problem is also linear [3] and, like the Couette problem, can be separated into the velocity slip and temperature jump problems. The problems of pure shear and velocity slip have been examined in [2].The temperature jump problem was examined in [4] for a model Boltzmann equation. For the linearized Boltzmann equation the problems noted above have been solved either by expanding the distribution function in orthogonal polynomials [5–7], which yields satisfactory results for small Knudsen numbers, or by the method of moments, with an approximation for the distribution function selected from physical considerations in the form of polynomials [8–10]. The solution presented below does not require any assumptions on the form of the distribution function.The concrete calculations were made for a molecular model that we call the Maxwell sphere model. It is assumed that the molecules collide like hard elastic spheres whose sections are inversely proportional to the relative velocity of the colliding molecules. A gas of these molecules is close to Maxwellian or to a gas consisting of pseudo-Maxwell molecules [3].     

Spherical expansion of vapor during evaporation of a droplet

The problem of the evaporation of a spherical particle is solved by a numerical finnite-difference method for the stationary and nonstationary cases on the basis of the generalized Krook kinetic equation [1]. Evaporation into a vacuum and into a flooded space are considered taking into account the reduction in size and cooling of the droplet. The minimum mass outflow is determined for stationary evaporation into a vacuum at small Knudsen numbers. The results are compared with those of other authors for both the spherical and plane problems. Most previous studies have used different approximations which reduce either to linearizing the problem [2, 3] or to use of the Hertz-Knudsen equation [4]. The inaccurate procedure of matching free molecular and diffusive flows at some distance from the surface of the droplet [5] is completely unsuitable in the absence of a neutral gas. Equations for the rate of growth of a droplet in a slightly supercooled vapor were obtained in [6] from a solution of the ellipsoidal kinetic model by the method of (expansion of) moments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 97–102, March–April, 1976.     

基于稀薄效应的微气体径向轴承稳态性能

针对微气体轴承给出参考努森数的定义,根据空气不同温度时的黏度,得到参考努森数的分布范围;考虑气体稀薄效应,给出基于Burgdorfer一阶滑移速度边界的微气体径向轴承润滑Reynolds方程的修正形式; 采用有限差分法求解修正的Reynolds方程,得到不同参考努森数$Kn_0$, 轴承数以及轴颈偏心率情况下轴承的压力分布、无量纲承载能力及偏位角. 数值分析表明:随气体稀薄程度的增强,气体径向轴承的压力明显降低,无量纲承载力降低,而轴承偏位角增大. 当偏心率小于0.6时,轴承偏位角变化平缓,受$Kn_0$数的影响不明显. 当轴承数较小时,气体稀薄程度对轴承的无量纲承载力、偏位角影响较小.      

Certain Aspects of an “Incompressible Fluid“ Model in Gas Dynamics

An “incompressible fluid” model in gas dynamics is developed in the linear approximation. Using the dissipative relaxation time as a characteristic scale, we arrive at another form of the dimensionless Boltzmann equation. In the limiting case of small Knudsen numbers an approximate solution is obtained in the form of a Hilbert multiple-scale asymptotic expansion. It is revealed that for slow, weakly nonisothermal processes the asymptotic expansion for the linearized Boltzmann equation leads in a first stage to equations for the velocity, pressure and temperature that do not contain the density (quasi-incompressible approximation). The density depends on the temperature and can, if necessary, be found from the equation of state. The next-approximation equations contain the Burnett effects, the velocity calculation being reduced to the general problem of finding a vector field from a given divergence and rotation. With reference to a simple case of the heating of a stationary gas in a half-space it is shown that the temperature establishment process is accompanied by gas flow from the wall.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 170–178.Original Russian Text Copyright © 2005 by Chekmarev.